Bought the game via Epic launcher and had the same problems.
This will happen when the installer don't create the savegame directories.
Have a try
Windows %USERPROFILE%\Saved Games\The Outer Worlds\
Microsoft Store %USERPROFILE%\Saved Games\The Outer Worlds Windows 10\<user-id>\
Linux (Proton) <Steam-folder>/steamapps/compatdata/578650/pfx/[Note 1]
https://www.pcgamingwiki.com/wiki/The_Outer_Worlds#Game_data